Equilibrium thermodynamics, Prigogine

I. Prigogine (Brussels) non-equilibrium thermodynamics, particularly the theory of dissipative structures. 

TNC.8. I. Prigogine and R. Balescu, Non-equilibrium thermodynamics, in Thermodynamics of Nuclear Materials, International Atomic Energy Agency, Vienna, 1962. 

TNC.75. G. Dewel, D. Kondepudi, and I. Prigogine, Chemistry far from equilibrium—Thermodynamics, Order and Chaos, New Chemistry, 1997. 

Prigogine, Etude Thermodynamique des Phenomenes Itreversibles, Desoer, Liege, 1947 S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover, New York, 1984. 

Gage, D. H., Schiffer, M., Kline, S. J. and Reynolds, W. C., The non-existence of a general thermodynamic variational principle, in "Non-Equilibrium Thermodynamics, Variational Techniques and Stability" (R. J. Donnelly, R. Herman, I. Prigogine, Eds.), p. 286. University of Chicago Press, Chicago (1966). 

We can specially show that the main principles of nonequilibrium thermodynamics (the Onsager relations, the Prigogine theorem, symmetry principle) and other theories of motion (for example, theory of dynamic systems, synergetics, thermodynamic analysis of chemical kinetics) are observed in the MEIS-based equilibrium modeling. In order to do that, we will derive these statements from the principles of equilibrium thermodynamics. 

First of all relying directly on the second law we will try to give the interpretation of the Prigogine theorem. Taking into account that the traditional variables of equilibrium thermodynamics are the parameters of state and, wishing to reveal the formalized relations between both thermodynamics, let us consider two situations sequentially (1) when some parameters of interaction that hinder the attainment of final equilibrium between the open subsystem and other parts of the isolated system that contains this subsystem are set (2) when flows are taken constant for the flow exchange between the open subsystem and the environment. It is obvious that both situations can be reduced to the case of fixing individual forces which is normally considered in the nonequilibrium thermodynamics. 

Ilya Prigogine (b. 1917 in Moscow, Russia) is Director of the International Solvay Institutes of Chemistry and Physics, Brussels, Belgium, and of the 1. Prigogine Center for Statistical Mechanics and Complex Systems, The University of Texas at Austin. He is a Belgian citizen. He received the Nobel Prize in Chemistry in 1977 for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures. ... 

Non-equilibrium thermodynamics was founded by Onsager. The theory was further elaborated by de Groot and Mazur and Prigogine. The theory is based on the hypothesis of local equilibrium a volume element in a non-equilibrium system is in local equilibrium when the normal thermodynamic relations apply to the element. Evidence is emerging that show that many systems of interest in the process industry are in local equilibrium by this criterion. " Onsager prescribed that each flux be connected to its conjugate force via the extensive variable that defines the flux. - ... 

Steady states are among the phenomena that non-equilibrium thermodynamics studies. When one is not too far from equilibrium it can be shown that the steady states are stable. On the other hand, when far from equilibrium, certain systems can make transitions to states exhibiting "dissipative structures." The theory of non-equilibrium developed by I. Prigogine, is quite general and has been applied to a wide range of phenomena. It is the aim of this lecture to introduce this field with a few examples. 

While Belousov was describing his e)q)eriments into oscillatory chemical reactions, Ilya Prigogine in Brussels was developing theoretical models of nonequilibrium thermodynamics and ended with the notion of "structure dissipative" for which he was awarded the 1977 Nobel Prize in Chemistry. The concept of "Dissipative Structure" is ejq)licitly mentioned in the Nobel quotation "The 1977 Nobel Prize in Chemistry has been awarded to Professor Ilya Prigogine, Brussels, for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures". In the first half of the 1950s, Glansdorff and Balescu defined with Prigogine the thermodynamic criteria necessary for oscillatory behavior in dissipative systems [7]. Nicohs and Lefever then applied these to models of autocatalytic reactions [8]. 

Dewel, Guy, DUip Kondepudi, and Ilya Prigogine. Chemistry Far from Equilibrium Thermodynamics, Order, and Chaos. In The New Chemistry, edited by Nona Hall, 440-446. Cambridge Cambridge University Press (2000). This chapter describes the mathematics of systems far removed fi-om equihbrium and the chemical consequences. 

Yet another important property of fractals which distinguishes them from traditional Euclidean objects is that at least three dimensions have to be determined, namely, d, the dimension of the enveloping Euclidean space, df, the fractal (Hausdorff) dimension, and d the spectral (fraction) dimension, which characterises the object connectivity. [For Euclidean spaces, d = d = d this allows Euclidean objects to be regarded as a specific ( degenerate ) case of fractal objects. Below we shall repeatedly encounter this statement] [27]. This means that two fractal dimensions, d( and d are needed to describe the structure of a fractal object (for example, a polymer) even when the d value is fixed. This situation corresponds to the statement of non-equilibrium thermodynamics according to which at least two parameters of order are required to describe thermodynamically nonequilibrium solids (polymers), for which the Prigogine-Defay criterion is not met [28, 29]. 

Non-equilibrium thermodynamics (de Groot and Mazur, 1984 Prigogine, 1977) affords an abstract, and therefore very general, foundation for understanding pattern formation. The abstract nature of this approach makes direct application to geochemical problems difficult, but non-equilibrium thermodynamics does provide a powerful conceptual basis for thinking about pattern formation. 

Prigogine is best known for extending the second law of thermodynamics to systems that are distant from equilibrium, showing that the formation of dissipative stmctures allows order to emerge from chaos in non-equilibrium systems. These stmctures have since been used to describe not only physical, chemical or biological phenomena, but also the growth of cities or the flow of traffic. Prigogine was awarded the Nobel Prize in Chemistry in 1977 "for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures". 

As far as equilibrium thermodynamics is regarded, we shall closely follow the lines of CALLEN s (1960) book which is particularly recommended for further reading An application of thermodynamics especially to chemical processes is presented in detail in the book of PRIGOGINE and DEFAY (1954). For an introduction into thermo-dynanjics of irreversible processes the interested reader is referred to the books of de GROOT (1951), de GROOT and MAZUR (1962) and KATCHALSKY and CURRAN (1967). 

Defay. R., and I. Prigogine Surface Tension and Adsorption, Longmans, London. 1966. de Groot, S.R., and P. Mazur Non-Equilibrium Thermodynamics, Dover Publications. Inc., New York, 1984. 

Gyarmati, I. (1970) Non-Equilibrium Thermodynamics, Springer, Berlin. Glansdorff, P. and Prigogine, I. (1971) Thermodynamic Theory of Structure, Stability and Fluctuations, John Wiley Sons, Ltd, New York. 

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